File:Solutions to Characteristic Polynomials of Degree 7 (in the complex plane).png

English: This image shows a 2D histogram on the complex plane. The intensity of each bin corresponds to the number of solutions from a given set of polynomials that fall in that area of the plane. To generate the set of polynomials, it is necessary to start with a set of 7x7 matrices. A matrix is, in this case, a rectangular array with entries set to either -1 or 1 that has seven rows and seven columns. There are over 560 trillion matrices in this set! Each matrix in this set has a characteristic polynomial that has 8 terms (7 with a variable, 1 constant). Setting these characteristic polynomials equal to zero and solving yields a large volume of complex numbers (imaginary numbers). This is the data used to fill in the 2D histogram! Each number is mapped to the bin in the correct location of the plane and that bin's count is increased by one. Displaying the histogram yields a sort of heat map, where intensity corresponds to the density of solutions in a given location on the plane. This heat map can be interpreted as a plot of the probability that a random matrix will have a characteristic polynomial with a solution in a certain location.